The spacetime of Einstein's relativity is 4-dimensional, but with a difference. So far we have assumed that the square of any vector x is a scalar, and that . For spacetime it is appropriate to make a different choice. We take the (+ - - -) metric usually preferred by physicists, with a basis for the spacetime algebra (STA) made up by the orthonormal vectors
These vectors obey the same algebraic relations as Dirac's -matrices, but our interpretation of them is not that of conventional relativistic quantum mechanics. We do not view these objects as the components of a strange vector of matrices, but (as with the Pauli matrices of 3-space) as four separate vectors, with a clear geometric meaning.
From this basis set of vectors we construct the 16 () geometric elements of the STA:
The time-like bivectors are isomorphic to the basis vectors of 3-dimensional space; in the STA they represent an orthonormal frame of vectors in space relative to the laboratory time vector [5,20]. The unit pseudoscalar of spacetime is defined as
which is indeed consistent with our earlier definition.
The geometric properties of spacetime are built into the mathematical language of the STA - it is the natural language of relativity. Equations written in the STA are invariant under passive coordinate transformations. For example, we can write the vector x in terms of its components as
These components depend on the frame and change under passive transformations, but the vector x is itself invariant. Conventional methods already make good use of scalar invariants in relativity, but much more power is available using the STA.
Active transformations are performed by rotors R, which are again even multivectors satisfying :
where the comprise a new frame of orthogonal vectors. Any rotor R can be written as
where is an arbitrary 6-component bivector ( and are relative vectors). When performing rotations in higher dimensions, a simple rotation is defined by a plane, and cannot be characterised by a rotation axis; it is an accident of 3-dimensional space that planes can be mapped to lines by the duality operation. Geometric algebra brings this out clearly by expressing a rotation directly in terms of the plane in which it takes place.
For the 4-dimensional generalisation of the gradient operator , we take account of the metric and write
where the are a reciprocal frame of vectors to the , defined via .
As an example of the use of STA, we consider electromagnetism, writing the electromagnetic field in terms of the 4-potential A as
The divergence term is zero in the Lorentz gauge. The field bivector F is expressed in terms of the more familiar electric and magnetic fields by making a space-time split in the frame:
Particularly striking is the fact that Maxwell's equations[20,27] can be written in the simple form
where J is the 4-current. Equation 5.11 contains all of Maxwell's equations because the operator is a vector and F is a bivector, so that the geometric product has both vector and trivector components. This trivector part is identically zero in the absence of magnetic charges. It is worth emphasising that this compact formula (5.11) is not just a trick of notation, because the operator is invertible. We can, therefore, solve for F:
The inverse operator is known to physicists in the guise of the Green's propagators of relativistic quantum mechanics. We return to this point in a companion paper, in which we demonstrate this inversion explicitly for diffraction theory.
It is possible here, as in three dimensions, to represent a relativistic quantum-mechanical spinor (a Dirac spinor) by the even subalgebra of the STA[7,24], which is 8-dimensional. We write this spinor as and, since contains only grade-0 and grade-4 terms, we decompose as
where R is a spacetime rotor. Thus, a relativistic spinor also contains an instruction to rotate - in this case to carry out a full Lorentz rotation. The monogenic equation in spacetime is simply
which, remarkably, is also the STA form of the massless Dirac equation. Furthermore, the inclusion of a mass term requires only a simple modification:
Figure 2: A charge moving in the observer's past light-cone
As a final example of the power of the STA in relativistic physics, we give a compact formula for the fields of a radiating charge. This derivation is as explicit as possible, in order to give readers new to the STA some feeling for its character, but nevertheless it is still as compact as any of the conventional treatments in the literature. Let a charge q move along a world-line defined by , where is proper time. An observer at spacetime position x receives an electromagnetic influence from the charge when it lies on that observer's past light-cone (see Figure 2). The vector
is the separation vector down the light-cone, joining the observer to this intersection point. We can take equation (5.16), augmented by the condition , to define a mapping from the spacetime position x to a value of the particle's proper time . In this sense, we can write , and treat as a scalar field. If the charge is at rest in the observer's frame we have
where r is the 3-space distance from the observer to the charge (taking c=1). For this simple case the 4-potential A is a pure electrostatic field, which we can write as
because . Generalising to an arbitrary velocity v for the charge, relative to the observer, gives
which is a particularly compact and clear form for the Liénard-Wiechert potential.
We now wish to differentiate the potential to find the Faraday bivector. This will involve some general results concerning differentiation in the STA, which we now set up; for further useful results see Chapter 2 of Hestenes & Sobczyk. Since the gradient operator is a vector we must take account of its commutation properties. Though it is evident that , we need also to deal with expressions such as , where a is a vector, and where the stars indicate that the operates only on x rather than a. The result is found by anticommuting the x past the a to give , and then differentiating this. Generalized to a grade-r multivector in an n-dimensional space, we have
Thus, in the example given above, . (See equation (4.20) for a 3-dimensional application of this result.)
We will also need to exploit the fact that the chain rule applies in the STA as in ordinary calculus, so that (for example)
since is a function of alone, and is the particle velocity. In equation (5.21) we use the convention that (in the absence of brackets or overstars) only operates on the object immediately to its right.
Armed with these results, we can now proceed quickly to the Faraday bivector. First, since
it follows that
As an aside, finding an explicit expression for confirms that the particle proper time can be treated as a scalar field - which is, perhaps, a surprising result. In the terminology of Wheeler & Feynman, such a function is called an `adjunct field', because it obviously carries no energy or charge, being merely a mathematical device for encoding information. We share the hope of Wheeler & Feynman that some of the paradoxes of classical and quantum electrodynamics, in particular the infinite self-energy of a point charge, might be avoidable by working with adjunct fields of this kind.
To differentiate A, we need . Using the results already established we have
which combine to give
Here, is the `acceleration bivector' of the particle:
The quantity is a pure bivector, because implies that . For more on the value of representing the acceleration in terms of a bivector, and the sense in which is the rest-frame component of a more general acceleration bivector, see Chapter 6 of Hestenes & Sobczyk.
The form of the Faraday bivector given by equation faraday is very revealing. It displays a clean split into a velocity term proportional to and a long-range radiation term proportional to . The first term is exactly the Coulomb field in the rest frame of the charge, and the radiation term,
is proportional to the rest-frame acceleration projected down the null-vector X.
Finally, we return to the subject of adjunct fields. Clearly X is an adjunct field, as was. It is easy to show that
In this expression for F we have expressed a physical field solely in terms of a derivative of an `information carrying' adjunct field. Expressions such as (5.32) and (5.33) (which we believe are new, and were derived independently by ourselves and David Hestenes) may be of further interest in the elaboration of Wheeler-Feynman type `action at a distance' ideas[28,29].